CHAPTER 11 Kinematics of Particles

11.1 INTRODUCTION TO DYNAMICS Galileo and Newton (Galileo’s experiments led to Newton’s laws) Galileo and Newton (Galileo’s experiments led to Newton’s laws) Kinematics – study of motion Kinematics – study of motion Kinetics – the study of what causes changes in motion Kinetics – the study of what causes changes in motion Dynamics is composed of kinematics and kinetics Dynamics is composed of kinematics and kinetics

RECTILINEAR MOTION OF PARTICLES

Velocity units would be in m/s, ft/s, etc. The instantaneous velocity is 11.2 POSITION, VELOCITY, AND ACCELERATION For linear motion x marks the position of an object. Position units would be m, ft, etc. Average velocity is

The average acceleration is The units of acceleration would be m/s 2, ft/s 2, etc. The instantaneous acceleration is

Notice One more derivative If v is a function of x, then

Consider the function x(m) t(s) v(m/s) a(m/s 2 ) t(s) Plotted t(s)

11.3 DETERMINATION OF THE MOTION OF A PARTICLE Three common classes of motion

with then get

Both can lead to or

11.4 UNIFORM RECTILINEAR MOTION

11.5 UNIFORMLY ACCELERATED RECTILINEAR MOTION Also

11.6 MOTION OF SEVERAL PARTICLES When independent particles move along the same line, independent equations exist for each. independent equations exist for each. Then one should use the same origin and time.

The relative velocity of B with respect to A The relative velocity of B with respect to A The relative position of B with respect to A Relative motion of two particles.

The relative acceleration of B with respect to A The relative acceleration of B with respect to A

Let’s look at some dependent motions.

A CD B EF G System has one degree of freedom since only one coordinate can be chosen independently. xAxA xBxB Let’s look at the relationships.

B System has 2 degrees of freedom. C A xAxA xCxC xBxB Let’s look at the relationships.

Skip this section. Skip this section GRAPHICAL SOLUTIONS OF RECTILINEAR-MOTION

Skip this section. Skip this section OTHER GRAPHICAL METHODS

11.9 POSITION VECTOR, VELOCITY, AND ACCELERATION CURVILINEAR MOTION OF PARTICLES x z y P P’P’ Let’s find the instantaneous velocity.

x z y P P’P’ x z y

x z y P P’P’ x z y x z y Note that the acceleration is not necessarily along the direction of the velocity.

11.10 DERIVATIVES OF VECTOR FUNCTIONS

Rate of Change of a Vector The rate of change of a vector is the same with respect to a fixed frame and with respect to a frame in translation.

11.11 RECTANGULAR COMPONENTS OF VELOCITY AND ACCELERATION

x z y x z y P

x z y

Velocity Components in Projectile Motion

x z y x’ z’ y’ O A B MOTION RELATIVE TO A FRAME IN TRANSLATION

Velocity is tangent to the path of a particle. Acceleration is not necessarily in the same direction. It is often convenient to express the acceleration in terms of components tangent and normal to the path of the particle TANGENTIAL AND NORMAL COMPONENTS

Plane Motion of a Particle O x y P P’P’

O x y P P’P’

Discuss changing radius of curvature for highway curves

Motion of a Particle in Space The equations are the same. O x y P P’P’ z

11.14 RADIAL AND TRANSVERSE COMPONENTS Plane Motion x y P

x y

Note

Extension to the Motion of a Particle in Space: Cylindrical Coordinates